nLab hyperbolic manifold

Idea

A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.

A hyperbolic manifold is a geodesically complete Riemannian manifold $(X,g)$ of constant sectional curvature $-1$.

Of particular interest are hyperbolic 3-manifolds.

Properties

Conformal flatness

Every hyperbolic manifold is a conformally flat manifold.

(e.g. Long-Reid 00, p. 4)

Zeta functions

There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.

Mostow rigidity theorem

The Mostow rigidity theorem states that every hyperbolic manifold of dimension $\geq 3$ and of finite volume is uniquely determined by its fundamental group.

Examples

• hyperbolic plane

• hyperbolic 3-manifold

  • hyperbolic 3-space

  • hyperbolic solid torus

    • BTZ black hole

    • thermal AdS3

Related concepts

A Riemannian manifold

• with zero sectional curvature is a Euclidean manifold?;

• with +1 sectional curvature is an elliptic manifold?

See also

• hyperbolic geometry

• hyperbolic link

• volume conjecture

• Selberg zeta function

• prime geodesic

• Bloch group

• M-theory on hyperbolic manifolds

References

Textbook accounts:

• John Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer 2006 (doi:10.1007/978-0-387-47322-2, pdf)

• Michael Kapovich, Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser Classics, Birkhäuser 2008 (doi:10.1007/978-0-8176-4913-5)

See also

• Darren D. Long, A. W. Reid, On the geometric boundaries of hyperbolic 4-manifolds (arXiv:math/0007197)

• Wikipedia, Hyperbolic manifold

• Wikipedia, Hyperbolic space